Batteries are important energy storage technologies in many industries and community settings. Users rely on batteries for reliable electrical power, and so for planning purposes they wish to know how much usage any particular battery may have; broadly speaking this concept is known as “state-of-charge”.
Manufacturing firms also care about state-of-charge, for example, in the development and deployment of battery-containing products that include state-of-charge indicators as a feature. Battery manufacturers are particularly concerned with establishing how fully charged a battery is before it is shipped, so that the customer can expect consistent high-level performance upon delivery. For example, the lead-acid battery is an electrochemical energy storage technology commonly used in systems all over the world, including backup power, electric vehicles, ignition systems, and renewable energy. The manufacturing process of these batteries involves a time and energy-intensive step known as ‘formation’, which electrochemically activates the batteries through a long electrical charge. In-situ measurement of the level of completeness of this charging process would provide more precise control over formation.
The Tafel equation is shown in Eq. (1), which describes the overvoltage (U−U0) of an electrochemical reaction generating an electric current. This equation breaks down at small overvoltages but holds true otherwise:
                              U          -                      U            0                          =                              U            Tafel            ′                    ⁢          log          ⁢                                          ⁢                      (                          I                              I                0                                      )                                              (        1        )            
In Eq. (1), I0 is the exchange current at the equilibrium voltage (U0) and U′Tafel is the Tafel slope. The Tafel slope is dependent on the gas constant (R), the temperature (T), the charge-transfer coefficient (α), the number of charges transferred (n), and the Faraday constant (F), as shown in Eq. (2). Values for U′Tafel have been established for many electrochemical reactions:
                              U          Tafel          ′                =                              ℛ            ⁢                                                  ⁢            T                                α            ⁢                                                  ⁢            n            ⁢                                                  ⁢            F                                              (        2        )            
In a battery, there is an unavoidable ohmic voltage drop whenever current is flowing, which is separate from the electrochemical overvoltage. This ohmic voltage drop is directly proportional to the current (I) and the ohmic resistance (RΩ). Thus for a battery, Eq. (1) can be expressed as Eq. (3):
                              U          -                      U            0                    -                      IR            Ω                          =                              U            Tafel            ′                    ⁢          log          ⁢                                          ⁢                      (                          I                              I                0                                      )                                              (        3        )            
The charging or discharging of a battery may also give rise to additional overpotentials from concentration gradients and crystallization processes, which can be grouped together as a “non-electrochemical overpotential” (η). This additional overpotential is included in Eq. (4):
                              U          -                      U            0                    -                      IR            Ω                    -          η                =                              U            Tafel            ′                    ⁢          log          ⁢                                          ⁢                      (                          I                              I                0                                      )                                              (        4        )            
If the goal is to determine Tafel slope for a given battery, then Eq. (4) can be rearranged into Eq. (5):
                              U          Tafel          ′                =                              U            -                          U              0                        -                          IR              Ω                        -            η                                              log              ⁢                                                          ⁢              I                        -                          log              ⁢                                                          ⁢                              I                0                                                                        (        5        )            
In practice, U, I, and RΩ can be measured during battery operation, but accurate determination of U0, I0, and η during operation is typically done through advanced modelling techniques or laborious scientific experimentations that are impractical for real-world applications.
Information regarding the state-of-charge of a battery, such as a lead-acid battery, is attainable using a variety of indicators including deep discharges, coulombic counting, electrolyte density, open-circuit voltage, loading response, ohmic resistance, electrochemical impedance spectroscopy, as well as modelling techniques such as Kalman filters, neural networks, and fuzzy logic. However, each of these methods has shortcomings of being too impractical, too inaccurate/unreliable, or too complex/expensive. There remains a need for a simple and accurate method for measurement of state-of-charge, providing value for both formation at the manufacturer and operation at the end-user.